Clearly x b0b1b2 b232 2e thus we have for any

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Unformatted text preview: . Clearly, x; = (b0:b1b2 : : :b23)2 2E : Thus we have, for any rounding mode, that jround(x) ; xj < 2;23 2E while for round to nearest jround(x) ; xj 2;24 2E : Similar results hold for double and extended precision, replacing 2;23 by 2;52 and 2;63 respectively, so that in general we have jround(x) ; xj < 2E (1) for any rounding mode and 1 jround(x) ; xj 2 2E for round to nearest. round towards zero, could the absolute rounding error be exactly equal to 2E ? For round to nearest, could the absolute rounding 1 error be exactly equal to 2 2E ? Exercise 10 For Exercise 11 Does (1) hold if x is subnormal, i.e. E = ;126 and b0 = 0? The presence of the factor 2E is inconvenient, so let us consider the relative rounding error associated with x, de ned to be = round(x) ; 1 = round(x) ; x : x x 14 Since for normalized numbers x = m 2E where m 1 (because b0 = 1) we have, for all rounding modes, E j j < 2E2 = : (2) In the case of round to nearest, we have jj Exercise 12 Does (2) hold if If not, how big could be? 1 2 2...
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